The Scientific The Scientific Method Method

and the Cycle of and the Cycle of ProofProof

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The Cycle of Proof:The Cycle of Proof:Daily Learning GoalDaily Learning Goal

The student will be The student will be able to identify able to identify and explain the and explain the steps of the cycle steps of the cycle of proof and apply of proof and apply them to examples them to examples from the history from the history of science.of science.

The following story describes how The following story describes how Eratosthenes, the librarian of the Eratosthenes, the librarian of the Library of Alexandria in the third Library of Alexandria in the third century B.C.E., was able to century B.C.E., was able to determine the size of the Earth.determine the size of the Earth.

First, read the story straight though. First, read the story straight though.

Then, look at the commentary, which Then, look at the commentary, which refers to the sections of the story refers to the sections of the story indicated by indicated by the superscript the superscript numbersnumbers11, to see how the scientific , to see how the scientific research described in the story research described in the story follows the scientific method, also follows the scientific method, also known as the cycle of proof.known as the cycle of proof.

Eratosthenes was a historian, Eratosthenes was a historian, geographer, and poet, but he geographer, and poet, but he was also notably a was also notably a mathematician and astronomer. mathematician and astronomer.

He was the first person to suggest He was the first person to suggest that an extra day be introduced that an extra day be introduced into the calendar every fourth into the calendar every fourth year to keep the calendar in line year to keep the calendar in line with the seasons – our leap with the seasons – our leap year.year.

In the library one day around 240 In the library one day around 240 B.C.E., he was reading a papyrus B.C.E., he was reading a papyrus book about Syene, a small town in book about Syene, a small town in the south of Egypt. One thing caught the south of Egypt. One thing caught his attention in particular. his attention in particular. At noon on At noon on midsummer day in Syene, vertical midsummer day in Syene, vertical sticks cast no shadow.sticks cast no shadow.

Also, the sun shone straight Also, the sun shone straight down a deep well, so that its down a deep well, so that its reflection could be seen by reflection could be seen by someone peering down the someone peering down the well into the water. well into the water. Eratosthenes was puzzled. If Eratosthenes was puzzled. If that had ever happened that had ever happened anywhere he'd been, he anywhere he'd been, he thought, he'd surely have thought, he'd surely have noticed it.noticed it.11

How could the sun, which Eratosthenes How could the sun, which Eratosthenes assumed was so far away that its rays assumed was so far away that its rays were parallel, cast different shadows in were parallel, cast different shadows in different locations on the same day at different locations on the same day at the same time? the same time? Eratosthenes reasoned Eratosthenes reasoned that it was the Earth that must be that it was the Earth that must be curved.curved.22

This idea of curvature, he realized, This idea of curvature, he realized, fitted in well with something else fitted in well with something else that people had always observed: that people had always observed: if a ship had just disappeared if a ship had just disappeared over the horizon, it could still be over the horizon, it could still be seen by someone at the top of a seen by someone at the top of a nearby hill. nearby hill.

It was for this reason that the It was for this reason that the lookout on a sailing ship was lookout on a sailing ship was always sent up to the “crow's always sent up to the “crow's nest” at the top of the mast, high nest” at the top of the mast, high above the deck, to sight land.above the deck, to sight land.

Eratosthenes predicted Eratosthenes predicted that a vertical stick in that a vertical stick in Alexandria would cast Alexandria would cast a shadow at noon on a shadow at noon on midsummer day, and midsummer day, and further, that the angle further, that the angle of the shadow could of the shadow could be used to determine be used to determine what fraction of the what fraction of the Earth's curvature was Earth's curvature was represented by the represented by the distance between distance between Alexandria and Alexandria and Syene.Syene.33

Eratosthenes set a man Eratosthenes set a man to pace the distance to pace the distance between Alexandria between Alexandria and Syene and and Syene and next next midsummer day, midsummer day, Eratosthenes set up Eratosthenes set up his vertical stick in his vertical stick in Alexandria and Alexandria and watched and watched and measured its measured its shadow.shadow.44

As the morning went on, the shadow As the morning went on, the shadow became shorter. At noon, it wasn't became shorter. At noon, it wasn't very long, but it was still there. very long, but it was still there. Eratosthenes measured the angle at Eratosthenes measured the angle at the top of the stick between the end the top of the stick between the end of the shadow and the stick. It was of the shadow and the stick. It was one-fiftieth of a complete circle – a one-fiftieth of a complete circle – a little more than little more than 77oo..55

So if the distance from Alexandria to So if the distance from Alexandria to Syene represents a fiftieth of a circle, Syene represents a fiftieth of a circle, then the distance around the world – then the distance around the world – the circumference – must be fifty the circumference – must be fifty times as far.times as far.

Since in our units, his estimate of the Since in our units, his estimate of the distance between the two cities distance between the two cities would be about 800 km, this would would be about 800 km, this would make the Earth's circumference 40 make the Earth's circumference 40 000 km, which we now know to be 000 km, which we now know to be approximately the right answer.approximately the right answer.

Not everyone in Eratosthenes' time agreed Not everyone in Eratosthenes' time agreed with him. The ancients found it hard to with him. The ancients found it hard to accept that their known world was such a accept that their known world was such a small fraction of the whole. Another Greek, small fraction of the whole. Another Greek, Poseidonius, made another, smaller Poseidonius, made another, smaller calculation of the circumference that was calculation of the circumference that was more accepted, and it was this calculation more accepted, and it was this calculation that Columbus used when planning his that Columbus used when planning his journey westward to India.journey westward to India.

If Columbus had known his journey If Columbus had known his journey was four times his estimate, he was four times his estimate, he probably would never have started probably would never have started out. It was not until Magellan's out. It was not until Magellan's expedition successfully expedition successfully circumnavigated the Earth that circumnavigated the Earth that Eratosthenes was finally proved Eratosthenes was finally proved correct.correct.

1. 1. The data baseThe data base. . The second paragraph of the story The second paragraph of the story

describes what was already describes what was already known before Eratosthenes known before Eratosthenes began to think about what it began to think about what it meant. A great deal of scientific meant. A great deal of scientific activity, not only then but also activity, not only then but also today, is concerned with today, is concerned with gathering this data base: gathering this data base: making and recording making and recording observations about the natural observations about the natural world.world.

A scientist or research team will A scientist or research team will always find out as much as possible always find out as much as possible about these observations that have about these observations that have already been made before beginning already been made before beginning any laboratory investigations. any laboratory investigations.

There are two main reasons for this: it There are two main reasons for this: it avoids wasting time and money avoids wasting time and money repeating things that have already repeating things that have already been tried, and examining the facts been tried, and examining the facts may suggest the kind of experiment may suggest the kind of experiment which is needed.which is needed.

2. 2. The hypothesisThe hypothesis. . From the data base, From the data base,

Eratosthenes arrived at a Eratosthenes arrived at a possible or tentative possible or tentative explanation for what was explanation for what was known. The explanation, in known. The explanation, in fact, turned out to be correct. fact, turned out to be correct. Even if it hadn't, though, it Even if it hadn't, though, it would still have provided a would still have provided a basis for the experiment that basis for the experiment that was being planned. This was being planned. This tentative explanation is tentative explanation is called the hypothesis.called the hypothesis.

Often, when scientists arrive Often, when scientists arrive at a hypothesis, they realize at a hypothesis, they realize that other facts that they that other facts that they already know make it more already know make it more likely to be correct. likely to be correct. Eratosthenes realized that Eratosthenes realized that observations made of ships observations made of ships or land appearing or or land appearing or disappearing over the disappearing over the horizon could also be horizon could also be explained by a curved Earth.explained by a curved Earth.

3. 3. The predictionThe prediction. . From the hypothesis, From the hypothesis,

scientists predict the scientists predict the outcome of a specific outcome of a specific experiment which they experiment which they have the expertise and have the expertise and the resources – the resources – equipment, time, equipment, time, manpower – to do.manpower – to do.

4. 4. The experimentThe experiment. . It may seem odd that the It may seem odd that the

prediction in the cycle of prediction in the cycle of proof comes ahead of proof comes ahead of designing the designing the experiment. In fact, both experiment. In fact, both will usually be in the will usually be in the scientist's mind at the scientist's mind at the same time.same time.

5. 5. VerificationVerification. . To verify something means to find out To verify something means to find out

whether it is true or not. In science, whether it is true or not. In science, the results of an experiment usually the results of an experiment usually show whether or not a prediction was show whether or not a prediction was correct. If, as in our example, things correct. If, as in our example, things turn out much as had been expected, turn out much as had been expected, scientists come away with scientists come away with strengthened confidence in the strengthened confidence in the original hypothesis.original hypothesis.

They still recognize that it They still recognize that it might not be correct, but might not be correct, but their work has increased the their work has increased the likelihood that it is. likelihood that it is.

Each experiment that tests the Each experiment that tests the hypothesis and verifies it hypothesis and verifies it strengthens it further. strengthens it further. Eratosthenes' prediction was Eratosthenes' prediction was further verified by the further verified by the circumnavigation of the globe circumnavigation of the globe and last century by and last century by observations from satellites.observations from satellites.

Often, though, a prediction turns out to Often, though, a prediction turns out to be wrong. The results of the be wrong. The results of the experiment are different from what experiment are different from what was expected. If this happens, the was expected. If this happens, the scientists will look closely at the way scientists will look closely at the way the experiment was set up and the experiment was set up and carried out. carried out.

(For example, if Eratosthenes' stick in (For example, if Eratosthenes' stick in Alexandria had not cast a shadow, he Alexandria had not cast a shadow, he would have had to check that it had would have had to check that it had been aligned correctly.)been aligned correctly.)

In the end, scientists have to be In the end, scientists have to be prepared to modify or discard a prepared to modify or discard a hypothesis. However, even hypothesis. However, even experiments in which it is discovered experiments in which it is discovered that the hypothesis must be that the hypothesis must be discarded are valuable since the size discarded are valuable since the size of the data base will have been of the data base will have been increased.increased.

QuestionQuestion: For each of the following statements, : For each of the following statements, determine which part of the cycle of proof (data determine which part of the cycle of proof (data base, hypothesis, prediction, experiment, or base, hypothesis, prediction, experiment, or verification) is demonstrated.verification) is demonstrated.

(a) The Earth is curved.(a) The Earth is curved.(b) The shadow of the Earth on the Moon is (b) The shadow of the Earth on the Moon is

observed to be curved.observed to be curved.(c) The next lunar eclipse is observed.(c) The next lunar eclipse is observed.(d) The Sun and the Moon are sometimes on exactly (d) The Sun and the Moon are sometimes on exactly

opposite sides of the Earth and the Earth casts a opposite sides of the Earth and the Earth casts a shadow on the Moon in an event called a lunar shadow on the Moon in an event called a lunar eclipse.eclipse.

(e) If the Earth is curved, the shape of the shadow it (e) If the Earth is curved, the shape of the shadow it casts on the Moon should be curved.casts on the Moon should be curved.

QuestionQuestion: For each of the following statements, : For each of the following statements, determine which part of the cycle of proof (data determine which part of the cycle of proof (data base, hypothesis, prediction, experiment, or base, hypothesis, prediction, experiment, or verification) is demonstrated.verification) is demonstrated.

(a) The Earth is curved. (a) The Earth is curved. hypothesishypothesis(b) The shadow of the Earth on the Moon is (b) The shadow of the Earth on the Moon is

observed to be curved.observed to be curved.(c) The next lunar eclipse is observed.(c) The next lunar eclipse is observed.(d) The Sun and the Moon are sometimes on exactly (d) The Sun and the Moon are sometimes on exactly

opposite sides of the Earth and the Earth casts a opposite sides of the Earth and the Earth casts a shadow on the Moon in an event called a lunar shadow on the Moon in an event called a lunar eclipse.eclipse.

(e) If the Earth is curved, the shape of the shadow it (e) If the Earth is curved, the shape of the shadow it casts on the Moon should be curved.casts on the Moon should be curved.

QuestionQuestion: For each of the following statements, : For each of the following statements, determine which part of the cycle of proof (data determine which part of the cycle of proof (data base, hypothesis, prediction, experiment, or base, hypothesis, prediction, experiment, or verification) is demonstrated.verification) is demonstrated.

(a) The Earth is curved. (a) The Earth is curved. hypothesishypothesis(b) The shadow of the Earth on the Moon is (b) The shadow of the Earth on the Moon is

observed to be curved. observed to be curved. verificationverification(c) The next lunar eclipse is observed.(c) The next lunar eclipse is observed.(d) The Sun and the Moon are sometimes on exactly (d) The Sun and the Moon are sometimes on exactly

opposite sides of the Earth and the Earth casts a opposite sides of the Earth and the Earth casts a shadow on the Moon in an event called a lunar shadow on the Moon in an event called a lunar eclipse.eclipse.

(e) If the Earth is curved, the shape of the shadow it (e) If the Earth is curved, the shape of the shadow it casts on the Moon should be curved.casts on the Moon should be curved.

QuestionQuestion: For each of the following statements, : For each of the following statements, determine which part of the cycle of proof (data determine which part of the cycle of proof (data base, hypothesis, prediction, experiment, or base, hypothesis, prediction, experiment, or verification) is demonstrated.verification) is demonstrated.

(a) The Earth is curved. (a) The Earth is curved. hypothesishypothesis(b) The shadow of the Earth on the Moon is (b) The shadow of the Earth on the Moon is

observed to be curved. observed to be curved. verificationverification(c) The next lunar eclipse is observed. (c) The next lunar eclipse is observed. experimentexperiment(d) The Sun and the Moon are sometimes on exactly (d) The Sun and the Moon are sometimes on exactly

opposite sides of the Earth and the Earth casts a opposite sides of the Earth and the Earth casts a shadow on the Moon in an event called a lunar shadow on the Moon in an event called a lunar eclipse.eclipse.

(e) If the Earth is curved, the shape of the shadow it (e) If the Earth is curved, the shape of the shadow it casts on the Moon should be curved.casts on the Moon should be curved.

QuestionQuestion: For each of the following statements, : For each of the following statements, determine which part of the cycle of proof (data determine which part of the cycle of proof (data base, hypothesis, prediction, experiment, or base, hypothesis, prediction, experiment, or verification) is demonstrated.verification) is demonstrated.

(a) The Earth is curved. (a) The Earth is curved. hypothesishypothesis(b) The shadow of the Earth on the Moon is (b) The shadow of the Earth on the Moon is

observed to be curved. observed to be curved. verificationverification(c) The next lunar eclipse is observed. (c) The next lunar eclipse is observed. experimentexperiment(d) The Sun and the Moon are sometimes on exactly (d) The Sun and the Moon are sometimes on exactly

opposite sides of the Earth and the Earth casts a opposite sides of the Earth and the Earth casts a shadow on the Moon in an event called a lunar shadow on the Moon in an event called a lunar eclipse. eclipse. data basedata base

(e) If the Earth is curved, the shape of the shadow it (e) If the Earth is curved, the shape of the shadow it casts on the Moon should be curved.casts on the Moon should be curved.

QuestionQuestion: For each of the following statements, : For each of the following statements, determine which part of the cycle of proof (data determine which part of the cycle of proof (data base, hypothesis, prediction, experiment, or base, hypothesis, prediction, experiment, or verification) is demonstrated.verification) is demonstrated.

(a) The Earth is curved. (a) The Earth is curved. hypothesishypothesis(b) The shadow of the Earth on the Moon is (b) The shadow of the Earth on the Moon is

observed to be curved. observed to be curved. verificationverification(c) The next lunar eclipse is observed. (c) The next lunar eclipse is observed. experimentexperiment(d) The Sun and the Moon are sometimes on exactly (d) The Sun and the Moon are sometimes on exactly

opposite sides of the Earth and the Earth casts a opposite sides of the Earth and the Earth casts a shadow on the Moon in an event called a lunar shadow on the Moon in an event called a lunar eclipse. eclipse. data basedata base

(e) If the Earth is curved, the shape of the shadow it (e) If the Earth is curved, the shape of the shadow it casts on the Moon should be curved. casts on the Moon should be curved. predictionprediction